L  B       .-IANGH 


tUnivetBtts  of  Gbicaao 

UC— NRLF         ..... ,,,,,  mi  mi 


THE  LEARNING  CURVE  EQUATION 


A  DISSERTATION 

SUBMITTED  TO   THE   FACULTY 

OF   THE   GRADUATE   SCHOOL   OF   ARTS   AND  LITERATURE 

IN   CANDIDACY  FOR   THE   DEGREE   OF 

DOCTOR   OF   PHILOSOPHY 

DEPARTMENT  OF  PSYCHOLOGY 


BY 

LOUIS  LEON  THURSTONE 


PSYCHOLOGICAL  MONOGRAPHS,  Vol.  XXVI,  Whole  No.  114 

PSYCHOLOGICAL  REVIEW  COMPANY 

PRINCETON,  NJ. 

1919 


EXCHANGE 


Ube  mniverstts  of  Cbicago 


THE  LEARNING  CURVE  EQUATION 


A  DISSERTATION 

SUBMITTED  TO  THE  FACULTY 

OF   THE   GRADUATE   SCHOOL   OF   ARTS  AND  LITERATURE 

IN   CANDIDACY  FOR  THE  DEGREE   OF 

DOCTOR  OF  PHILOSOPHY 

DEPARTMENT  OF  PSYCHOLOGY 


BY 

LOUIS  LEON  THURSTONE 


PSYCHOLOGICAL  MONOGRAPHS,  Vol.  XXVI,  Whole  No.  114 

PSYCHOLOGICAL  REVIEW  COMPANY 

PRINCETON,  N  J. 

1919 


ACKNOWLEDGMENT 

The  subjects  for  this  study  in  learning  were  students  at  the 
Duff  Business  School  in  Pittsburgh.  I  wish  to  acknowledge  the 
interest  and  cooperation  of  Messrs.  Spangler  and  Johnson  and 
of  Miss  Wilson  for  making  the  necessary  rearrangements  at  the 
Duff  School  for  this  investigation.  I  wish  to  acknowledge  also 
the  advice  and  interest  of  Dean  J.  R.  Angell  under  whose  direc- 
tion this  study  has  been  carried  out  and  the  many  favors  of  Pro- 
fessor W.  V.  Bingham  without  which  the  work  could  not  have 
been  completed.  I  am  also  thankful  to  Professor  Dorweiler  of 
the  Carnegie  Institute  of  Technology  for  his  kind  advice  on 
some  of  the  mathematical  portions  of  the  study. 


THE  LEARNING  CURVE  EQUATION 
Introduction 

I  Methods  of  investigation  PAGE 

1 )  Verbal  statement 2 

2)  Correlation  statistics 2 

3)  Empirical  equations 6 

a)  Method  of  inspection 7 

b)  Method  of  regression  equation 7 

c)  Method  of  least  squares 9 

4)  Rational  equations    10 

II    The  learning  curve  equation 

1 )  Purpose  of  the  equation 1 1 

2)  The  equation   12 

a)  Method  of  least  squares 14 

Case  i :  When  it  passes  through  the  origin 
Case  2 :  When  it  does  not  pass  through  the 
origin 

b)  Method  of  inspection 17 

c)  Method  of  three  equidistant  points 18 

3)  Interpretation  of  learning  curve  constants 19 

4)  The  coordinates 19 

a)  Speed-time  curve 

b)  Time-amount  curve 

c)  Time-time  curve 

d)  Speed-amount  curve 

5)  Initial  positive  acceleration 22 

6)  Other  possible  equations 23 

III  Typewriter  learning 

1 )  The  subjects 26 

2)  Coordinates  of  curves  for  typewriting 28 

3)  Learning  coefficients  for  typewriting 29 

4)  The  findings 31 

a)  Writing  speed   31 

b)  The  errors  33 

c)  The  variability 34 

IV  Summary 

1 )  Forms  of  the  learning  curve 35 

2)  Outline  for  calculating  the  learning  coefficients. . .   37 

3)  Typewriter  learning 


INTRODUCTION 

The  present  investigation  is  essentially  an  attempt  to  devise 
a  statistical  method  for  treating  learning  data.  Part  I  is  a  dis- 
cussion of  correlation  methods  and  empirical  and  rational  equa- 
tions. Part  II  is  a  description  of  the  learning  curve  equation 
and  its  interpretation.  Part  III  is  a  discussion  of  the  application 
of  the  learning  curve  equation  to  typewriter  learning.  Part  IV 
is  a  summary. 

Learning  curves  are  usually  very  erratic  and  for  this  reason 
it  is  necessary  to  study  the  general  trend  of  numerous  observa- 
tions instead  of  the  variable  individual  observations.  The 
methods  to  be  discussed  often  make  it  possible  to  obtain  co- 
efficients which  express  the  characteristics  of  a  subject's  learn- 
ing based  on  all  the  observations  and  in  such  a  manner  that  all 
observations  are  as  far  as  possible  equally  weighted.  Quantita- 
tive methods  in  psychology  are  far  in  advance  of  our  control 
over  the  things  measured,  and  consequently  we  make  ourselves 
subject  to  ridicule  when  refined  correlation  statistics  are  applied 
to  measures  which  are  obviously  crude.  We  shall  therefore  dis- 
cuss not  only  the  more  refined  statistical  procedure  for  the  learn- 
ing curve  but  also  some  readily  applied  methods  which  are 
adaptable  in  the  study  of  more  or  less  erratic  learning  data. 
Even  though  refined  technique  is  available  we  should  select  the 
quantitative  methods  for  any  particular  study  so  as  to  keep  a 
fair  balance  between  the  certainty  of  our  measures  and  the  statis- 
tical niceties  by  which  we  treat  them. 

For  the  benefit  of  any  reader  who  wishes  to  apply  the  statis- 
tical methods  to  be  described  for  his  own  learning  data  I  wish 
to  call  attention  to  the  first  two  sections  of  the  summary  in  which 
will  be  found  an  outline  describing  the  detailed  procedure  in 
calculating  the  learning  coefficients. 

In  applying  these  methods  of  learning  curve  analysis  one 
should  be  fully  aware  of  their  limitations.  They  are  not  ap- 
plicable to  the  following  conditions  of  learning:  i)  when  trial 


2  L.  L.  THURSTONE 

and  error  learning  is  mixed  with  generalizations  such  as  in  puzzle 
solving;  2)  when  the  learning  is  so  erratic  that  it  fails  to  show 
continuity;  3)  when  the  learning  process  has  not  been  carried 
far  enough  to  reveal  the  nature  of  the  function;  which  is  often 
the  case  with  apparently  linear  learning  curves;  4)  when  the 
learning  curve  is  not  plotted  in  the  speed-amount  form;  5)  when 
the  learning  curve  fails  to  show  diminishing  returns  with  prac- 
tice; 6)  when  the  units  of  formal  practice  are  variable  in  the 
different  stages  of  learning  (learning  measured  on  different 
successive  scales  can  not  be  treated  as  a  continuous  function) ; 
7)  when  the  wrong  responses  are  eliminated  by  ideational  learn- 
ing without  giving  any  objective  scores  during  the  process  of 
elimination.  Such  learning  curves  have  the  same  appearance 
as  those  which  contain  generalizations. 

1)  VERBAL  STATEMENT  OF  RELATIONSHIP 

Our  present  problem  concerns  the  relationship  between  prac- 
tice and  attainment  in  learning.  When  an  observer  notes  as  an 
element  in  common  experience  that  attainment  increases  as  prac- 
tice increases,  he  may  generalize  by  verbally  asserting  a  positive 
relation  between  the  two  variables.  The  verbal  generalization 
is  so  common  that  it  is  embodied  in  what  we  call  common  sense. 
Thus  we  expect  without  further  verification  that  twenty  hours 
of  practice  in  a  complex  function  will  yield  higher  attainment 
than  ten  hours  of  practice  under  roughly  similar  conditions,  but 
uncontrolled  observation  does  not  tell  us  how  much  higher. 

2)  THE  CORRELATION  COEFFICIENT  AS  AN  EXPRESSION  OF  RE- 

LATIONSHIP 

It  is  possible  to  express  by  a  single  number  the  degree  of  rela- 
tionship between  two  variables.  This  is  what  one  attempts  to 
do  by  a  correlation  coefficient.  The  Pearson  coefficient  of  corre- 
lation is  so  derived  that  when  its  value  is  unity  the  two  variables 
have  perfect  concomitance.  When  its  value  is  — i.  the  two 
variables  have  perfect  inverse  relationship,  a  rise  in  one  of  the 
variables  being  always  associated  with  a  proportional  decrease 
in  the  other.  A  zero  correlation  establishes  the  fact  of  entire 


THE  LEARNING  CURVE  EQUATION  3 

absence  of  relationship  within  the  conditions  of  the  experiment. 

A  correlation  coefficient  considerably  less  than  unity  may  be 
explained  in  at  least  four  different  ways:  i)  the  observations 
themselves  may  be  so  inaccurate  as  to  obscure  the  relationship; 
2)  the  two  variables  may  be  related  through  a  common  third 
variable  which,  if  not  controlled  or  kept  constant,  plays  havoc 
with  the  experiment;  3)  the  regression  may  be  non-linear  in 
which  case  the  Pearson  coefficient,  r,  is  almost  meaningless;* 
4)  the  two  variables  may  be  intrinsically  independent.  Psycho- 
logical experimentation  rarely  yields  correlations  over  0.85  be- 
cause of  the  inaccuracy  of  psychological  measures.  When  a 
correlation  coefficient  turns  out  to  be  0.95  or  above  an  empirical 
equation  may  properly  be  substituted  for  the  correlation  methods. 

In  the  interpretation  of  a  correlation  coefficient  one  should  be 
careful  to  note  that  while  a  high  correlation  coefficient  does  indi- 
cate a  relation  between  the  variables  under  the  conditions  of  the 
experiment,  a  low  coefficient  does  not  indicate  the  absence  of 
relationship  between  the  variables.  The  first,  second,  or  third 
factors  enumerated  in  the  preceding  paragraph  may  be  responsi- 
ble for  a  low  coefficient  when  a  high  relation  really  exists.  It  is 
perhaps  rare  that  a  correlation  is  calculated  with  psychological 
data  which  is  not  grossly  affected  by  all  three  of  these  factors. 

When  one  variable  is  immediately  contingent  on  one  or  more 
other  variables  it  is  advisable  to  use  the  method  of  partial  corre- 
lation to  establish  the  relation.  Thus  the  volume  of  a  box  is 
contingent  on  the  three  variables,  length,  width,  and  depth.  Now, 
if  a  heterogeneous  collection  of  wooden  boxes  were  to  be  meas- 
ured as  to  all  four  of  these  attributes  and  the  correlation  co- 
efficient between  volume  and  length  determined,  it  would  un- 
doubtedly turn  out  to  be  positive  and  significant  because  long 
boxes  are  usually  more  voluminous  than  short  boxes.  But  the 

*The  term  regression  was  introduced  by  Galton  in  connection  with  his 
statistical  studies  in  the  heredity  of  stature.  It  is  the  equation  of  the  best 
fitting  line  for  a  series  of  paired  observations  of  two  variables.  The  use 
of  the  term  seems  to  be  restricted  to  data  of  considerable  dispersion  and  is 
not  used  for  those  observations  in  the  exact  sciences  which  do  not  involve 
serious  scatter.  The  term  linear  regression  refers  to  the  equation  of  a  re- 
gression line  which  is  straight,  as  contrasted  with  non-linear  regressions 
which  are  curved.  See  Yule,  "Textbook  in  Theory  of  Statistics,"  p.  176 
and  references  2  and  3  on  p.  188. 


4  L.  L.  THURSTONE 

coefficient  would  not  be  unity  because  of  the  two  other  variables 
which  were  left  out  of  consideration.  This  would  illustrate 
case  2  in  the  preceding  paragraph.  We  may  distinguish  two 
methods  of  handling  this  type  of  relation,  a)  We  may  control 
the  extraneous  variables  by  keeping  them  constant  while  meas- 
uring the  two  variables  with  which  we  are  immediately  con- 
cerned. This  is  the  customary  procedure  in  the  physical  sciences. 
Thus  when  verifying  Boyle's  Law  we  keep  the  temperature  of 
the  gas  constant,  but  when  verifying  Charles'  Law  we  keep  the 
pressure  constant.  Except  for  these  precautions  neither  of  the 
two  laws  would  be  observed.  In  the  biological  sciences  we  do 
not  have  such  ready  control  over  the  extraneous  variables  and 
the  best  we  can  do  is  to  measure  them  also,  while  allowing  them 
to  vary  at  will,  b)  In  these  cases  we  use  a  second  method, 
namely,  partial  correlation.  Thus  if  we  calculate  the  partial 
correlation  coefficient  between  volume  and  length  of  the  collec- 
tion of  wooden  boxes,  with  the  depth  and  width  accounted  for, 
we  obtain  a  higher  coefficient  than  when  the  two  extraneous 
variables  were  ignored.  The  advantage  of  the  partial  correla- 
tion method  is  that  it  enables  us  to  control  the  extraneous  vari- 
ables analytically  without  having  any  physical  control  over  them. 
There  are  some  limitations  of  the  correlation  methods  which 
every  experimenter  should  keep  in  mind  in  order  to  guard 
against  erroneous  conclusions.  One  of  the  limitations  of  the 
partial  correlation  method  is  that  it  assumes  the  combined  effect 
of  the  several  independent  variables  on  the  dependent  variable 
to  be  additive,  a  condition  which  rarely  obtains.  This  limitation 
is  much  more  serious  than  the  inadequacy  of  the  correlation 
coefficient  for  non-linear  regressions.  A  non-linear  regression 
can  usually  be  rectified  by  one  of  the  algebraic  artifices  used  in 
connection  with  empirical  equations,  but  the  assumption  that  the 
several  independent  variables  produce  their  effect  on  the  de- 
pendent variable  in  an  additive  manner  can  not  be  handled  by 
any  predetermined  statistical  method.  Thus,  returning  to  the 
box  illustration,  the  best  measure  we  could  obtain  by  the  method 
of  partial  correlation  is  expressed  in  the  form 

v  =  ktd  +  k2w  +  k3l.  i 


THE  LEARNING  CURVE  EQUATION  5 

But  the  true  formula  for  the  volume  takes  the  form 

v  =  k .  d .  w .  1.  2 

Now,  this  type  of  relation  is  not  revealed  by  the  partial  cor- 
relation method,  nor  is  the  method  adequate  for  any  of  the 
thousands  of  ways  in  which  the  several  variables  may  combine 
except  the  additive  one. 

If  we  are  not  content  with  merely  stating  in  quantitative  form 
the  degree  of  relationship  between  the  two  variables  but  wish  to 
formulate  a  method  of  prediction,  we  use  the  regression  equa- 
tion. This  equation  is  derived  from  the  Pearson  coefficient 
which  merely  states  in  numerical  form  the  degree  of  relation- 
ship between  the  two  variables.  It  places  the  information  at 
our  immediate  command  for  the  purpose  of  prediction.  Given 
one  of  the  unknowns,  the  other  can  be  found  either  from  the 
regression  lines  or  from  the  regression  equation  which  is  simply 
an  algebraic  description  of  the  regression  line. 

It  is  quite  conceivable  that  a  low  value  of  attribute  A  may 
be  associated  with  either  high  or  low  value  of  attribute  B, 
whereas  a  high  value  of  attribute  A  may  be  associated  with 
only  high  values  of  attribute  B.  Similarly  a  low  value  of  at- 
tribute B  may  be  associated  with  low  values  of  attribute  A  only, 
whereas  high  values  of  B  are  associated  with  either  high  or 
low  values  of  attribute  A.  Whenever  such  conditions  obtain 
the  Pearson  correlation  coefficient  is  inadequate  to  express  the 
complete  relationship.  In  these  cases,  called  non-linear  regres- 
sions, it  is  advisable  to  calculate  another  kind  of  coefficient  which 
is  called  the  eta  coefficient,  >?,*  or  correlation  ratio.  The  sig- 
nificance of  the  correlation  ratio  may  perhaps  be  made  more 
apparent  by  the  analogue  All  dogs  are  quadrupeds  but  all  quad- 
rupeds are  not  dogs.  What  would  correspond  to  the  correlation 
ratio  of  dogs  on  quadrupeds  would  be  very  high  because  all 
dogs  are  quadrupeds.  But  the  correlation  ratio  of  quadrupeds 
on  dogs  would  be  low,  for  only  a  few  of  the  quadrupeds  are 
dogs.  The  Pearson  correlation  coefficient  for  a  relation  of  this 
type  would  be  positive  but  low. 

*For  a  brief  statement  giving  the  derivation  of  the  correlation  ratio,  see 
Yule,  p.  204. 


6  L.  L.  THURSTONE 

There  are  a  number  of  algebraic  artifices  by  means  of  which 
a  non-linear  regression  may  be  rectified.  The  value  of  such 
devices  becomes  apparent  when  it  is  considered  that  such  other- 
wise exceedingly  useful  tools  as  are  available  in  the  correlation 
methods  are  inapplicable  as  long  as  the  regressions  are  non- 
linear. The  investigator  must  rely  on  his  own  ingenuity  in  rec- 
tifying a  non-linear  regression.  Some  of  these  methods  will  be 
considered  in  connection  with  the  learning  curve  equation. 

3)  EMPIRICAL  EQUATIONS 

Every  equation  can  be  represented  by  a  line  in  a  diagram  and 
practically  every  line  encountered  in  quantitative  experimental 
work  can  be  represented  by  an  equation.  Thus  the  regression 
equation  is  only  an  algebraic  way  of  describing  the  regression 
line  of  a  scatter  diagram,  or,  putting  it  the  other  way,  the  re- 
gression line  is  a  graphical  description  of  the  regression  equa- 
tion. Each  tells  the  same  story  in  its  respective  language. 

An  empirical  equation  is  an  equation  selected  to  fit  a  given 
set  of  data.  The  observations  give  us  the  diagram  and  if  we 
find  an  equation  whose  line  coincides  with  the  general  trend  of 
the  observations,  it  may  be  used  interchangeably  with  the  dia- 
gram for  predicting  one  of  the  attributes  when  the  other  is 
given.  When  the  observations  indicate  a  linear  relation  we  can 
derive  the  corresponding  equation  with  very  little  trouble,  but 
when  the  observations  fall  along  a  curve  and  when  they  are 
badly  scattered  the  finding  of  the  most  representative  empirical 
equation  sometimes  taxes  the  investigator's  ingenuity.  I  shall 
describe  the  routine  steps  in  determining  the  empirical  equation 
for  a  linear  relation  by  means  of  an  example  and  will  show  that 
it  turns  out  to  be  identical  with  the  regression  equation  for  the 
same  data. 

Figure  i  is  a  diagram  of  the  relation  between  two  hypo- 
thetical variables  X  and  Y.  Each  of  the  small  circles  repre- 
sents a  hypothetical  observation;  the  solid  line  represents  the 
general  trend  of  the  observations.  This  line  may  be  used  for 
the  purpose  of  prediction.  Thus  if  we  know  that  on  a  certain 
occasion  attribute  X  had  a  numerical  value  of  9,  the  attribute  Y 


THE  LEARNING  CURVE  EQUATION  7 

must  have  been  very  close  to  the  value  63,  as  read  from  the 
chart.  Our  problem  now  is  to  describe  this  line  algebraically 
so  that  the  prediction  may  be  made  by  means  of  a  formula  in- 
stead of  by  the  diagram. 

a)  Method  of  inspection.    This  procedure  is  the  simplest  but 
it  can  only  be  applied  when  the  relation  is  close,  as  it  is  in  the 
present  illustration.    We  first  indicate  the  observations  by  small 
circles  or  dots  on  the  diagram.    Then  we  draw  by  inspection  the 
best  fitting  straight  line  through  the  general  trend  of  the  ob- 
servations.   The  equation  of  a  straight  line  always  takes  the  form 

Y  =  a  +  b.X  3 

The  y-intercept  is  30.8  and  it  is  the  constant  a.  The  slope  of 
the  line  is  3.56  and  it  is  the  constant  b.  Hence  the  equation  for 
the  line  is 

Y  =  30.8  +  3.56X  4 

This  equation  may  be  used  interchangeably  with  the  diagram 
in  predicting  one  of  the  attributes  when  the  other  is  known. 
The  procedure  is  so  simple  and  direct  that  it  would  be  uni- 
versally used,  were  it  not  for  the  fact  that  when  the  observa- 
tions scatter  badly,  it  is  difficult  to  draw  the  best  fitting  straight 
line  by  inspection.  Moreover,  the  same  line  can  not  properly 
be  used  in  predicting  X  from  Y  as  in  predicting  Y  from  X 
when  the  data  are  scattered.  In  these  cases  we  have  recourse 
to  two  other  types  of  procedure,  namely,  the  method  of  the  re- 
gression equation  and  the  method  of  least  squares. 

b)  Method  of  regression  equation.    In  figure  2  we  have  repre- 
sented a  hypothetical  set  of  data  which  are  quite  scattered.    By 
the  usual  correlation  methods  we  obtain  the  following  constants : 

r     =  +  0.59 

C7X      =  4.58 

"y    =          3-03 

n  =  50. 
mx  =  10. 
my  =  8. 

The  regression  equation  with  two  variables  for  predicting  X 
from  Y  takes  the  form 


L.  L.  THURSTONE 


x  =  IW  —  y 

^ 

in  which  x  and  y  are  deviations  of  X  and  Y  from  their  respective 
means.  Rewriting  this  equation  in  terms  of  the  variables  X  and 
Y  instead  of  in  terms  of  the  deviations  from  their  means,  we  have 

o-j. 
X  —  mI  =  rxy  —  (Y  —  my)  6 

ar 

in  which  mx  and  my  are  the  arithmetic  means  of  X  and  Y  re- 
spectively. 

Substituting  the  numerical  values  into  equation  6  we  have 


—  8) 


3-03 
and  simplifying,  we  obtain 

X  =  o.89Y  +  2.88  8 

This  is  the  regression  equation  ready  for  use.  By  means  of 
it  we  predict  X  when  the  value  of  Y  is  known.  When  the  data 
are  not  seriously  scattered  it  is  safe  to  fit  the  line  by  inspection 
and  determine  the  empirical  equation  by  the  shorter  method, 
but  with  the  data  of  figure  2  the  regression  lines  can  hardly  be 
judged  by  inspection. 

By  analogy  the  regression  equation  for  predicting  Y  from  X 
takes  the  form 


which  when  stated  in  terms  of  the  variables  instead  of  in  terms 
of  the  deviations  from  their  respective  means  becomes 

CTy 

Y  —  my  =  rxy  —  (X<  —  mx)  10 

^x 

Substituting  and  simplifying,  as  before,  we  get  as  a  numerical 
statement  of  the  relation  that 

Y  =  0.39X4-  4.1  ii 

which  is  ready  for  use  in  predicting  the  most  probable  value  of 
Y  for  a  known  value  of  X. 

These  regression  equations  may  be  obtained  with  less  arith- 
metical labor,  particularly  if  only  one  of  the  regression  equations 
is  needed,  by  the  method  of  least  squares. 


THE  LEARNING  CURVE  EQUATION  9 

c)  Method  of  least  squares.  The  method  of  least  squares  is 
an  aid  in  finding  the  best  fitting  straight  lines  for  representing 
a  series  of  observations.  It  can  be  applied  also  to  curves  but 
that  often  leads  to  awkward  mathematical  maneuvering.  The 
method  of  least  squares  gives  a  line  such  that  the  sum  of  the 
squares  of  the  deviations  of  the  independent  variable  from  the 
regression  line  is  a  minimum.  It  is  the  best  fitting  straight  line 
for  the  observations.  The  method  gives  the  numerical  values 
of  the  constants  a  and  b  in  the  equation 

X  =  a+-b-Y  3 

which  represents  any  straight  line  and  by  means  of  which  we 
can  predict  the  most  probable  value  of  X  from  a  given  value 
of  Y.  The  application  of  the  method  consists  in  solving  the  two 
following  formulae: 

)-SX.Y)  —  2Y2).SX) 


—  n-S(Y») 


b-  —  n.28(X.Y) 


—  n-S(Y«) 

Substituting  the  appropriate  sums  from  the  data  we  find  that 
a  =  2.9  and  b  =  o.89.  Hence  the  equation  of  the  best  fitting 
line  for  predicting  X  from  Y  is 

X  =  2.90  +  0.89Y  14 

It  should  be  noted  that  this  equation,  as  determined  by  the 
method  of  least  squares,  is  identical  with  the  regression  equation 
8  which  was  determined  by  correlation  methods. 

By  analogy,  the  equation  for  the  straight  line  by  which  the 
most  probable  value  of  Y  may  be  determined  from  a  known 
value  of  X  is 

Y  =  c  +  d-X  15 

The  constants  c  and  d  may  be  determined  from  the  original  ob- 
servations by  the  formulae: 


_ 

[5(X)]2  —  n-2(X2) 

d=        *(X)'*(Y)—  n-*(X-Y) 


io  L.  L.  THURSTONE 

Substituting  the  numerical  value  of  the  data  from  figure  2  into 
equations  15  and  16  we  obtain:  0  =  4.09,  and  b  =  o.39.  Sub- 
stituting these  numerical  values  into  15  we  get 

Y  =  4.09  +  0.39X  17 

by  means  of  which  we  may  predict  the  most  probable  value  of 
Y  from  a  known  value  of  X. 

It  can  be  shown  readily  that  when  the  regression  equations 
have  been  determined  by  the  method  of  least  squares  the  Pearson 
coefficient  of  correlation  is  expressed  by  the  relation 

r=Vb-d*  18 

in  which  b  and  d  are  identical  with  the  regression  coefficients. 

The  above  calculations  make  it  apparent  that  the  method  of 
least  squares  gives  us  a  pair  of  regression  lines  which  are  iden- 
tical with  those  obtained  by  the  correlation  methods.  For  the 
purpose  of  stating  quantitatively  the  degree  of  relationship  be- 
tween two  variables  it  is  desirable  to  calculate  the  correlation 
coefficient.  When  the  regression  equation  is  of  primary  interest 
it  can  be  calculated  to  advantage  by  the  method  of  least  squares 
particularly  if  only  one  of  the  regressions  is  needed. 
4)  RATIONAL  EQUATIONS 

A  rational  equation  is  derived  from  known  relations  and  is 
verified  by  experimental  observation.  From  a  philosophic  point 
of  view  one  may  argue  that  all  equations  used  by  science  are  in 
the  last  analysis  empirical  but  in  practice  there  is  a  far  cry  be- 
tween fitting  an  empirical  equation  to  a  series  of  observations 
and  the  ability  to  predict  the  observed  relations  on  the  basis  of 
a  rational  equation. 

Psychology  has  very  few  bona  fide  laws  on  which  we  can 
build  a  system  of  quantitative  prediction  and  control.  Thus 
practically  all  we  can  do  with  the  problem  of  learning  is  to  ob- 
serve the  function  and  describe  it.  The  present  attempt  is  to  de- 
scribe it  quantitatively  by  an  empirical  equation.  Some  day  we 
shall  possess  in  psychology  a  coordinated  system  of  really  work- 
able concepts  with  objective  reference  by  which  we  may  be  able 
to  predict  and  control  at  least  certain  aspects  of  behavior  by 
rational  equations  or  their  equivalents. 

*See  Yule,  p.  203. 


II.  THE  LEARNING  CURVE  EQUATION 

i )  PURPOSE  OF  THE  EQUATION 

When  the  learning  function  for  a  simple  coordination  pro- 
ceeds undisturbed  by  external  or  internal  distraction  it  usually 
follows  a  law  of  diminishing  returns.  In  the  majority  of  learn- 
ing curves  the  amount  of  attainment  gained  per  unit  of  practice 
decreases  as  practice  increases.  Exceptions  to  this  tendency  are 
found  in  studying  the  learning  of  complex  processes  such  as  a 
foreign  language,  and  when  successive  generalizations  are  in- 
volved such  as  puzzle  solving  and  the  like.  These  exceptions 
sometimes  take  the  form  of  a  positive  acceleration  at  the  initial 
stage  of  the  learning,  plateaus  during  the  course  of  learning 
and  erratic  advance  of  attainment.  But  these  irregularities 
should  not  stand  in  the  way  of  an  attempt  to  express  the  learn- 
ing function  as  a  law  provided  that  we  do  it  with  due  conserva- 
tism in  its  interpretation.  All  we  can  hope  to  do  in  thus  ex- 
pressing the  learning  function  is  to  formulate  what  can  with 
considerable  certainty  be  considered  as  the  typical  relation  be- 
tween practice  and  attainment. 

Besides  giving  the  satisfaction  of  formulating  the  relation 
between  practice  and  attainment,  the  use  of  an  equation  for  this 
relation  enables  one  to  predict  the  limit  of  practice  before  it 
has  been  attained,  provided  that  the  learning  follows  the  law 
of  diminishing  returns.  It  also  enables  one  to  differentiate  for 
various  purposes  the  rate  of  learning  from  the  limit  of  practice 
since  these  two  attributes  are  undoubtedly  independent.  It  en- 
ables us  to  state  how  much  preceding  practice  the  subject  has 
experienced  under  the  assumption  that  the  learning  function 
followed  the  same  law  before  and  after  the  formal  measure- 
ments. Another  use  for  which  the  equation  can  be  of  service 
is  in  the  analysis  of  the  relation  between  the  variability  in  learn- 
ing and  other  mental  attributes.  The  problems  of  formal  disci- 
pline may  be  investigated  by  ascertaining  whether  a  succession 


12  L.  L.  THURSTONE 

of  learning  processes,  all  of  the  same  type,  yields  any  rise  in 
the  limit  of  practice,  or  a  higher  rate  of  learning,  or  a  greater 
consistency  of  learning  in  the  successive  learning  processes. 
Some  of  these  coefficients  may  be  more  susceptible  than  others 
to  modification  by  successive  repetition  of  the  same  type  of  learn- 
ing. This  would  in  reality  be  studying  the  problems  of  learning 
how  to  learn.  All  questions  of  transfer  of  training  may  be  in- 
vestigated by  the  learning  equation  and  the  transfer  effect  may 
be  differentiated  into  its  psychological  components.  Thus,  con- 
tinued practice  in  learning  poetry  may  show  no  rise  of  the  prac- 
tice limit,  but  a  considerable  rise  in  the  rate  at  which  that  limit 
is  approached  and  in  a  decrease  of  the  variability  of  the  learning. 
Relearning  may  be  found  to  approach  the  same  limit  of  practice 
as  the  initial  learning  but  it  may  proceed  at  a  higher  rate,  and 
this  rate  can  be  stated  as  a  coefficient  which  is  independent  of 
the  amount  of  previous  practice  in  each  learning  process.  The 
laws  of  forgetting  are  expressible  in  terms  quite  similar  to  those 
here  used  for  the  learning  function.  It  is  not  at  all  unlikely  that 
these  coefficients  may  come  to  be  significant  in  individual  psy- 
chology quite  apart  from  their  immediate  utility  as  descriptive 
attributes  of  the  learning  function.  The  preceding  remarks 
have,  I  hope,  justified  my  attempt  to  devise  a  method  for  in- 
vestigating the  learning,  memory,  and  forgetting  functions. 
2)  THE  EQUATION 

After  experimenting  with  some  forty  different  equations  on 
published  learning  curves  I  have  selected  a  form  of  the  hyperbola 
as  being  for  practical  purposes  the  most  available.  It  takes  the 
form 

L-X 

Y  = 19 

X  +  R 

in  which 

Y  =  attainment  in  terms  of  the  number  of  successful  acts  per 

unit  time. 
X  =  formal  practice  in  terms  of  the  total  number  of  practice 

acts  since  the  beginning  of  formal  practice. 
L  =  Limit  of  practice  in  terms  of  attainment  units. 
R  =  Rate  of  learning  which  indicates  the  relative  rapidity 

with  which  the  limit  of  practice  is  being  approached.    It 


THE  LEARNING  CURVE  EQUATION  13 

is  numerically  high  for  a  low  rate  of  approach  and  nu- 
merically low  for  a  high  rate  of  approach. 

Equation  19  represents  a  learning  curve  which  passes  through 
the  origin,  i.e.,  it  starts  with  a  zero  score  at  zero  formal  practice. 
The  majority  of  learning  curves  start  with  some  finite  score 
even  at  the  initial  performance.  For  learning  curves  which  do 
not  pass  through  the  origin,  the  equation  becomes 

_L(X±P)_ 

(X+P)+R 

in  which  P  =  equivalent  previous  practice  in  terms  of  formal 
practice  units. 

Figure  3  represents  the  learning  curve  for  subject  No.  23  in 
the  group  of  fifty-one  typewriter  students  to  be  discussed  in  a 
later  section.  This  curve  is  plotted  between  attainment,  Y,  in 
terms  of  the  number  of  words  written  in  a  four  minute  test 
given  weekly  for  seven  months,  and  formal  practice  (X)  in 
terms  of  the  total  number  of  pages  written  since  entering  the 
course.  We  shall  call  this  type  of  curve  the  speed-amount  curve 
to  distinguish  it  from  other  ways  of  plotting  the  same  data. 

Equation  19  may  be  rectified  as  follows: 

Y==  L-x 

X  +  R 
XY  +  R-Y  =  LX 

X  +  R=±L(^)  21 

This  equation  is  linear  if  X/Y  is  plotted  against  X.     Similarly 
equation  21  may  be  rectified  when  written  in  the  form 

X+  (R  +  P)  =L  (X+-P)  22 

which  becomes  linear  when  (X  +  P)/Y  is  plotted  against  X. 

When  so  rectified,  the  constants  L,  R,  and  P  may  be  deter- 
mined by  several  different  methods,  the  choice  between  which 
depends  on  the  scatter  of  the  data,  the  desired  accuracy,  and  the 
number  of  curves  one  has  to  calculate.  We  shall  describe  four 
methods  of  calculating  the  coefficients. 


14  L.  L.  THURSTONE 

a)  Method  of  least  squares 

Case  i:  when  the  learning  curve  passes  through  the  origin: 
Arrange  the  data  as  in  table  i.  Calculate  X/Y  and  tabulate. 
Plot  X/Y  against  X  as  in  figure  4.  For  convenience  we  shall  call 

Y  =  Z  23 

The  reader  will  notice  that  the  learning  data  as  plotted  in  figure 
4  falls  practically  in  a  straight  line  whereas  the  same  data  in 
figure  3  takes  the  typical  learning  curve  form. 

The  equation  for  the  best  fitting  straight  line  of  figure  4  can 
be  represented  by  the  equation 

Z  =  c  +  d-X  24 

The  constants  c  and  d  are  determined  from  the  table  of  data  by 
the  formulae: 


d  =  -  .  -  26 


which  are  simply  the  least  square  formulae  (15)  and  (16)  re- 
written for  X  and  Z.  Substituting  the  proper  sums,  we  have 
0  =  0.42,  and  d  =  0.0041.  Hence  the  equation  for  X  and  Z 
becomes 

Z  =  0.42  +  0.0041  X  27 

which  by  replacing  X/Y  for  Z  and  transposing  becomes 

Y=     244X  28 

X+I02 

in  which  the  predicted  limit  of  practice,  L,  is  244  words  in  four 
minutes,  and  the  rate  of  learning,  R,  is  102.  The  constants  L 
and  R  may  also  be  determined  by  the  relations 


Z  =  i/d 

Plotting  equation  (28)  we  obtain  the  solid  line  in  figure  3. 
It  will  be  noticed  that  this  curve  fits  quite  well  the  general  trend 
of  the  observations  which  are  indicated  by  the  small  circles.  This 
method  of  stating  algebraically  the  relation  between  practice  and 


THE  LEARNING  CURVE  EQUATION  15 

attainment  is  of  course  not  applicable  unless  the  speed-amount 
curve  for  the  data  takes  the  typical  hyperbolic  form. 

Case  2:  when  the  learning  curve  does  not  pass  through  the 
origin.  When  the  learning  curve  does  not  pass  through  the  ori- 
gin it  can  be  rectified  by  slightly  different  procedure.  We  shall 
take  as  an  illustration  the  combined  curve  for  a  group  of  fifty- 
one  subjects  studying  typewriting.  Figure  5  represents  the 
average  speed  of  typewriting  against  the  total  number  of  pages 
written  since  entering  the  course.  It  is  seen  to  be  a  fairly  smooth 
and  regular  curve. 

The  equation  for  the  learning  curve  which  does  not  pass 
through  the  origin  is 

Y  =  .    L<X  +  P) 

(X  +  P)+R 

or,  if  we  call 

P  +  R  =  K  32 

for  convenience,  we  have,  instead  of  equation  (20) 

Y  =     L(X  +  P)  33 

X  +  K 

This  equation  can  be  rectified  as  follows :  When  X  =  o,  and 
Y  =  Y±,  Y±  being  the  initial  attainment  score, 

L-P 
Y1  =  —  34 

and  hence  equation  (33)  becomes 

-^L-L-^-    -K  K 

Y-Y~    ^Y-Y, 

This  equation  is  linear  if  XY/(Y — Y±)  is  plotted  against 
X/(Y — Yj),  in  which  case  L  is  the  multiplying  constant  and 
K  is  the  additive  constant. 

Plotting  the  data  represented  in  figure  5  in  this  manner  we 
obtain  figure  6  in  which  the  learning  data  appear  as  a  straight 
line.  This  line  may  be  represented  by  the  equation 

S  =  a  +  b.T  36 

in  which 

x  X.Y 

" 


16  L.  L.  THURSTONE 

The  numerical  values  of  a  and  b  may  be  determined  by  the  fol- 
lowing least  square  formulae  which  are  identical  with  equations 
12  and  /J,  except  for  the  analogous  notation. 

S 


—  n-(T2) 

^^     a(T).S(S)-n.a(S.T) 

[2(T)]2  —  n.2(T2) 

Substituting  the  proper  sums  we  find  that  a  =  —  148.  and 
b  =  216.  Hence 

S  =  148.  +  2i6.T  39 

which  is  the  equation  of  the  solid  line  in  figure  6.  This  equation 
may  be  transposed  into  the  original  form  of  equation  20,  or  we 
may  write  it  in  that  form  directly  by  the  following  relations: 

a=K 
b  =  L 

P  =     a'Yl 
L 

R  =  K  —  P 

All  of  the  constants  K,  L,  Y,  P,  and  R,  are  positive  when  ap- 
plied to  learning  curves.  It  should  be  noted  that  Yx  is  a  repre- 
sentative original  score  determined  by  projecting  the  learning 
curve  back  to  the  y-axis.  In  figure  5  the  actually  observed 
initial  score  was  used  since  it  is  continuous  with  the  rest  of  the 
data.  But  it  is  occasionally  necessary  to  select  a  representative 
initial  score  since  Yx  is  weighted  in  this  procedure  more  than 
any  of  the  other  points.  The  numerical  values  of  these  con- 
stants for  the  data  of  figure  5  are  as  follows  : 

L==2i6. 

P=    19. 

R=i33- 

Substituting  these  constants  in  equation  jj  we  have 

2i6.(X+i9.) 

Y  -  -  ,  AQ 

X  +  148. 

which  when  plotted  becomes  the  solid  line  of  figure  5.  The 
reader  will  notice  that  this  equation,  as  represented  by  the  solid 


THE  LEARNING  CURVE  EQUATION  17 

line  in  figure  5,  is  a  beautiful  fit  for  the  data,  and  it  justifies 
our  use  of  equations  ip  and  20  to  represent  the  hyperbolic  form 
of  learning  curve. 

The  predicted  limit  of  practice  L  which  is  216.  words  in  four 
minutes,  is  of  course  based  on  the  assumption  that  the  learning 
curve  would  continue  as  uniformly  beyond  the  measurements 
as  it  did  during  the  measurements.  This  limitation  must  be  kept 
in  mind  and  we  shall  therefore  differentiate  between  the  pre- 
dicted limit  and  a  limit  of  practice  which  has  been  practically 
attained.  The  equivalent  previous  practice  (P)  is  ip  pages 
which  we  may  interpret  as  the  average  number  of  pages  of  type- 
writing to  which  the  previous  general  experience  of  our  subjects 
was  equivalent.  This  interpretation  of  the  constant  P  is  also 
limited  by  the  assumption  that  the  unmeasured  learning  function 
followed  the  law  which  the  measurements  reveal.  One  circum- 
stance which  bears  out  this  assumption  is  that  those  learning 
.curves  which  actually  do  pass  through  the  origin  and  which  do 
not  show  positive  acceleration  usually  follow  this  curve  law 
when  the  coordinates  are  properly  chosen.  The  curve  of  figure 
5  does  not  pass  through  the  origin  but  this  is  explainable  by  the 
fact  that  a  person  who  has  never  touched  a  typewriter  will  in 
four  minutes  make  some  finite  score  even  though  handicapped 
by  using  the  hunt  and  punch  method. 
b)  Method  of  inspection 

When  the  observations  fall  very  nearly  in  a  straight  line  as 
they  do  in  figure  6  it  is  hardly  necessary  to  plough  through  the 
arithmetical  labor  involved  in  evaluating  the  constants  a  and  b 
of  equation  36  by  the  method  of  least  squares  unless  one  has 
ready  access  to  a  calculating  machine.     After  plotting  figure  6 
one  may  draw  at  sight  the  best  fitting  straight  line  through  the 
general  trend  of  the  data  and  evaluate  the  constants  from  any 
of  the  following  relations: 
y-intercept  =  a  =  K 
x-intercept  = 
slope  =  L 


L 
=  K  —  P 


i8  L.  L.  THURSTONE 

By  this  graphical  procedure  much  labor  is  saved  in  calculating 
the  learning  curve  constants  and  the  method  is  identical  with 
the  preceding  in  principle. 
c)  Method  of  three  equidistant  points 

The  learning  coefficients  may  be  determined  from  three  se- 
lected points  with  less  labor  than  when  all  the  observations  are 
taken  into  account.  These  three  points  should  be  so  selected  that 
they  represent  the  general  trend  of  the  learning  curve. 

Let  the  three  selected  points  be  denoted  XjY^;  X2Y2;  and 
X3Y3.  Let  Xx  be  zero,  X3  the  total  amount  of  practice  and  X2 
the  midpoint  between  X!  and  X3.  Let  the  Y-values  be  the  most 
representative  ordinates  to  the  curve.  Then 

X3  =  2-X2  41 

By  substituting  these  values  into  equation  20,  transposing  and 
simplifying,  we  obtain 

X3(Y2-X8) 

42 


Y1-2Y2 

3  +  K)-Y1-K 


43 


Y±-K 

p=-ir 

R  =  K  —  P  45 

From  these  relations  we  may  determine  the  numerical  values  of 
the  learning  coefficients  in  terms  of  the  three  equidistant  points. 
When  the  curve  passes  through  the  origin  both  Xx  and  Y± 
are  zero.  The  coefficients  may  then  be  determined  by  the  fol- 
lowing somewhat  simpler  relations  : 

X3(Y2-Y3) 

K  =  .  -  46 

Y3-2-Y2 

Y3(X3  +  K) 

~~~ 


R  =  K 

P  is  zero  because  when  the  initial  score  is  zero  the  equivalent 
previous  practice  is  zero. 


THE  LEARNING  CURVE  EQUATION  ig 

3)  INTERPRETATION  OF  LEARNING  CONSTANTS 

We  have  seen  that  the  learning  curve  equation  20  fits  very 
well  the  learning  data  to  which  we  have  applied  it.  In  order 
to  bring  out  the  interpretation  of  the  learning  coefficients  we 
shall  compare  several  learning  curves  with  high  and  low  nu- 
merical values  of  the  coefficients. 

In  figure  7  we  have  two  hypothetical  learning  curves  with  dif- 
ferent physiological  limits  but  with  identical  rates  of  approach. 
Figure  8  represents  two  hypothetical  learning  curves,  both  ap- 
proaching the  same  limit  of  practice,  one  at  a  high  rate  and  the 
other  at  a  low  rate.  Figure  9  represents  two  hypothetical  learn- 
ing curves  with  same  limit  of  practice,  and  with  the  same  rate 
of  approach,  but  differing  in  the  amount  of  previous  practice. 
Curve  A  represents  forty  units  of  previous  practice  while  curve 
B  represents  no  previous  practice.  The  two  curves  are  identical 
in  shape,  the  only  difference  between  them  being  that  curve  B 
is  forty  x-units  to  the  right  of  curve  A.  The  same  interpreta- 
tion would  be  reached  if  the  two  curves  were  superimposed  and 
the  formal  practice  measurements  started  at  the  origin  for  curve 
B  and  after  forty  practice  units  for  A. 

4)  THE  COORDINATES  FOR  LEARNING  CURVES 

So  far  we  have  considered  learning  curves  plotted  only  be- 
tween the  coordinates  X  (total  number  of  practice  acts  since 
the  beginning  of  practice)  and  Y  (the  number  of  successful  acts 
per  unit  time).  Learning  curves  have,  however,  been  plotted 
with  various  units  for  the  coordinates  and  we  shall  consider 
several  of  these  together  with  some  inferences  that  may  be  drawn 
from  the  translation  of  learning  data  from  one  system  of  units 
to  another. 

The  speed-amount  curve  is  the  name  we  shall  use  to  designate 
the  form  of  learning  curve  we  have  been  considering.  It  is 
plotted  as  speed,  Y,  against  amount  of  practice,  X.  It  may  be 
represented  by  equations  19  and  20  when  it  reveals  the  typical 
hyperbolic  form. 

The  time-amount  curve  is  plotted  as  time,  t,  per  unit  amount 
of  work  against  total  amount  of  work,  X,  since  the  beginning 


20  L.  L.  THURSTONE 

of  practice.    It  is  evident  that  the  ordinates  of  this  type  of  curve 
will  be  proportional  to  the  reciprocals  of  the  speed-amount  curve 
for  the  same  data.    Hence  we  may  define  t  as 
C 

t==Y 

where  t  is  the  time  per  unit  amount  of  work  and  C  is  a  constant. 
Limiting  ourselves  to  the  curves  of  diminishing  returns  we  have, 
as  the  equation  of  the  time-amount  curve 

C(X  +  K) 

t  = 49 

L(X  +  P) 

The  constant  C  is  only  significant  in  translating  learning 
curves  from  one  form  to  the  other.  Applying  the  equation  di- 
rectly to  learning  data  the  constant  C  may  be  dropped.  In  that 
case 

X  +  R 
t  =  — , 5o 

L(X  +  P) 

This  equation  may  be  rectified  by  the  procedure  previously  out- 
lined for  equations  ip  and  20.    When  P  =  zero,  we  have 
_X  +  R 

:  L-X 

which  can  be  rectified  by  plotting  tX  against  X. 

In  order  to  determine  whether  equations  50  and  51  really  fit 
the  time-amount  curve  throughout  its  range  I  have  given  a  long 
substitution  test  to  one  of  my  students.  He  took  the  test  seven- 
teen times,  once  a  day,  and  reached  what  is  for  all  practical 
purposes  a  practice  limit.  The  time-amount  curve  for  this  learn- 
ing test  is  represented  in  figure  u.  In  figure  12  I  have  rectified 
the  data  by  plotting  the  products  tX  against  X.  The  reader 
will  notice  that  the  speed-amount  curve  is  hyperbolic.  It  is  quite 
gratifying  that  the  learning  records  for  an  individual  subject 
follow  the  hyperbolic  law  so  closely.  In  order  to  avoid  erratic 
scores  from  individual  subjects  it  is  absolutely  essential  that 
they  work  under  uniform  conditions  with  a  minimum  amount 
of  distraction.  The  student  whose  substitution  learning  is  repre- 
sented in  figures  n  and  12  took  the  test  once  a  day  only  and 


THE  LEARNING  CURVE  EQUATION  21 

always  at  I  P.  M.     The  test  consisted  in  making  six  hundred 
substitutions  at  each  sitting. 

The  time-time  curve  is  the  learning  curve  plotted  between  the 
time,  t,  per  unit  amount  of  work  and  the  total  time,  T,  devoted 
to  practice.  An  empirical  equation  may  be  derived  for  this  type 
of  curve  from  the  assumed  hyperbolic  form  of  the  speed-amount 
curve.  The  total  time  is  the  summation  2t-dx  for  the  whole 
period  of  learning.  Hence 

T=J*t-dx  $2 

But  from  equation  50 

X  +  K 

50 


L(X  +  P) 
and  hence 

X  +  K 


dX 


L(X  +  P) 

which  may  be  written 
i 

~L   J  '       L        J  X  +  P 

Integrating,  we  obtain 

X        K    —  P 
T  =  —  +  — log  (X  +  P)  +  d  53 

-L/  L-J 

which  gives  the  equivalent  total  time  T  in  terms  of  X.  Stating 
X  explicitly  from  equation  50  and  substituting  in  equation  53 
gives  the  desired  relation  between  T  and  t  as 

K  — t-L-P       K  — P          K— P 

~  (L-t— i)L  L       °g    L-t— I 

While  this  equation  does  give  us  a  relation  between  T  and  t  as 
derived  from  the  hyperbolic  speed-amount  curve  it  is  too  un- 
wieldy to  be  practically  feasible.  We  are  hardly  justified  in 
using  so  complex  an  empirical  equation  for  learning  data. 

The  speed-time  curve  is  plotted  between  the  speed  Y  (number 
of  successful  acts  per  unit  time)  and  the  total  amount  of  time, 
T,  devoted  to  practice.  An  equation  between  Y  and  T  may  be 
derived  by  stating  X  explicitly  from  equation  20  in  terms  of  Y 
and  substituting  this  for  X  in  equation  53  which  gives 


22  L.  L.  THURSTONE 

P_K        K—  P  K—  P 


i0g(L-Y)+C2    55 

While  this  equation  is  too  cumbersome  for  extensive  use  it 
serves  one  very  interesting  function  in  that  it  sheds  light  on  the 
question  of  positive  acceleration  in  learning  curves. 

5)  INITIAL  POSITIVE  ACCELERATION  IN  THE  SPEED-TIME  CURVE 

Equation  55  represents  the  speed-time  curve.  It  may  be  sim- 
plified by  letting 

A  =  P—  -  KandB=  K~"P  55a 

when  it  becomes 


The  first  derivative  with  respect  to  Y  is 

dT  ABB 

--- 


dY  (Y  —  L)2        Y        L  —  Y 

The  second  derivative  is 

d2T  =       2A  B  B 

dY2  "     (Y  —  L)3     Y2       (Lr—  Y)2 
which  when  simplified  becomes 

d2T         B.(L)2.(3.Y  —  L) 

dY2  "          Y2-(L  —  Y)3 
since  A  =  —  BL  from  equation  55a 
Equating  the  second  derivative  to  zero,  we  have: 

B.(L)2.(3.Y-L)    ^Q 

Y2-(L  —  Y)3 
which  is  true  when  Y  has  the  value  L/3. 

This  demonstrates  the  presence  of  a  point  of  inflection  in 
equation  55  at  the  value  L/3  for  Y.  The  psychological  signifi- 
cance of  this  relation  may  be  stated  as  follows: 

The  learning  curve  in  the  speed-time  form  must  necessarily 
have  an  initial  positive  acceleration  which  changes  to  a  negative 
acceleration  when  the  attainment  has  reached  one-third  of  the 
limit  of  practice.  This  conclusion  is  contingent  on  the  assump- 


THE  LEARNING  CURVE  EQUATION  23 

tion  that  the  learning  curve  in  the  speed-amount  form  is  hyper- 
bolic, an  assumption  which  has  been  empirically  shown  to  be  safe 
for  the  majority  of  learning  curves.  As  has  already  been  said, 
the  speed-amount  curve  is  usually  hyperbolic  but  not  always. 
These  assertions  regarding  the  speed-time  curve  are  not  applica- 
ble when  the  speed-amount  curve  for  the  same  data  is  not  hyper- 
bolic. The  positive  acceleration  can  not,  of  course,  be  obtained 
when  the  initial  score  is  greater  than  one-third  of  the  practice 
limit.  It  can  only  be  observed  when  the  initial  score  is  less  than 
one- third  of  the  practice  limit. 

In  order  to  test  empirically  the  above  rinding  with  regard  to 
initial  positive  acceleration  I  have  plotted  in  figure  13  the  average 
typewriting  speed  for  fifty-one  subjects  against  weeks  of  prac- 
tice (the  speed-time  curve)  instead  of  against  total  number  of 
pages  written  (the  speed-amount  curve).  The  average  practice 
limit  for  this  group  has  already  been  found  to  be  216  words  in 
four  minutes  according  to  the  speed-amount  curve  for  the  same 
data.  The  reader  will  notice  the  initial  positive  acceleration  fol- 
lowed by  negative  acceleration,  and  also  that  the  point  of  transi- 
tion from  positive  to  negative  acceleration  takes  place  at  a 
writing  speed  of  about  seventy  words  in  four  minutes,  as  it 
should  do  according  to  our  analysis  of  the  speed-time  curve.  If 
this  rinding  will  stand  the  test  of  further  experimentation  it  is 
obviously  of  considerable  diagnostic  value  for  the  psychologist 
who  can  by  means  of  it  predict  the  practice  limit  when  attain- 
ment reaches  one- third  of  its  limit.  The  limitation  in  the  use 
of  this  relation  is  mainly  in  the  erratic  improvement  in  complex 
coordinations  which  are  learned  under  variable  conditions  of 
distraction  and  in  the  occasional  deviations  from  the  typical 
hyperbolic  form  of  the  speed-amount  curve. 

6)  OTHER  POSSIBLE  EQUATIONS 

Before  closing  the  discussion  on  the  learning  curve  equation 
as  such  it  might  not  be  out  of  place  to  mention  a  few  of  the  other 
equations  which  I  have  tried  to  use  for  learning  data.  These 
will  not  be  of  interest  to  the  general  reader  but  may  be  of  in- 
terest to  those  who  wish  to  try  their  hand  at  other  empirical 
equations  for  the  learning  function. 


24  L.  L.  THURSTONE 

One  of  these  equations  for  the  speed-amount  curve  is 

Y  =  L[i-    -i-]  60 

where  £  is  the  Naperian  base  or  some  other  constant.  This  equa- 
tion can  not  readily  be  rectified  except  by  trying  successive  values 
for  L.  When  the  proper  value  for  L  is  found  it  can  be  rectified 
when  written  in  the  form 

log  (L  —  Y)  =  log  L  —  a-X-log  e  61 

by  plotting  log(L  —  Y)  against  X.  If  the  curve  does  not  pass 
through  the  origin  equation  60  becomes 

Y  =  L[i  --  -  -  ]  62 

ea(x+p) 

which  is  rectified  if  the  proper  numerical  values  of  L  and  P  are 
found  by  writing  it  in  the  form 

log(L—  Y)  =  log  L—  a(X+P)log  e  63 

and  plotting  log(L  —  Y)  against  (X+P).  This  equation  gives 
a  fair  approximation  to  the  speed-amount  curve  but  it  does  not 
fit  nearly  as  well  as  the  hyperbolic  form  previously  considered. 
It  can  be  rectified  graphically  by  plotting  Y-increments  against  X 
but  this  procedure  is  not  feasible  unless  the  individual  observa- 
tions are  more  consistent  than  they  usually  are  for  learning  data. 
The  constants  L  and  P  can  also  be  determined  graphically  from 
three  selected  points.  If  X-^,  X2Y2,  and  X3Y3  be  three  points 
on  the  curve,  equidistant  on  the  axis  of  abscissae,  then  the  two 
lines  X3Y2  ;  X2Yi  and  X3Y3  ;  X2Y2  will  intersect  in  a  point  which 
is  on  the  asymptote  parallel  to  the  axis  of  abscissae,  thus  deter- 
mining the  constant  L  graphically.  This  equation  gives  a  fair 
approximation  to  the  speed-amount  curve  but  it  does  not  fit 
nearly  as  well  as  the  hyperbolic  form  previously  considered. 
Another  equation  which  gives  a  fair  approximation  to  the 

learning  curve  is 

c 

Y  =  L(B*)  64 

in  which  B  and  C  are  constants.  It  can  be  rectified  by  writing 
it  in  the  form 


logY  =  logL  —       logB  65 

X 


THE  LEARNING  CURVE  EQUATION  25 

and  plotting  logY  against  i/X.  It  has  the  advantage  of  sim- 
plicity and  it  can  be  used  to  represent  an  initial  positive  accelera- 
tion. But  as  far  as  I  have  been  able  to  determine  the  constant  L 
does  not  agree  as  well  with  observed  values  as  the  hyperbolic 
form. 

One  could  perhaps  write  an  indefinite  number  of  exponential, 
trigonometric  and  other  functions  to  represent  the  learning  curve 
but  as  long  as  the  simple  equation  20  with  its  various  trans- 
formations fits  the  data,  and  as  long  as  we  do  not  have  the  basis 
for  a  rational  equation  for  learning  I  have  been  content  to 
abide  by  it. 


TYPEWRITER  LEARNING 

i)  THE  SUBJECTS 

Eighty-three  students  at  the  Duff  Business  School  in  Pitts- 
burgh took  one  four  minute  typewriter  test  once  a  week  during 
the  school  year  1916-17.  The  tests  were  begun  in  September 
and  continued  until  the  middle  of  April.  The  subjects  practiced 
two  hours  of  school  schedule  time  every  day,  five  days  a  week. 
No  tests  are  available  for  the  first  three  weeks  of  practice  be- 
cause teachers  of  typewriting  who  use  the  touch  system  prefer 
not  to  give  tests  from  straight  copy  until  the  mechanism  of  the 
typewriter  and  the  key  board  have  been  mastered.  This  takes 
from  three  to  seven  weeks,  depending  on  the  maturity,  adapta- 
bility and  industry  of  the  students.  Practically  all  of  the  sub- 
jects had  finished  the  grammar  school,  a  number  of  them  had 
completed  one  or  two  years  of  high  school,  and  several  had  fin- 
ished a  four  year  high  school  course.  Their  average  age  was 
about  seventeen  years.  In  order  to  obtain  an  initial  typewriting 
score  I  asked  ten  of  my  students  who  had  never  touched  a  type- 
writer to  take  a  four-minute  test.  The  average  score  for  this 
group  was  27  words  in  four  minutes  and  this  is  used  with  the 
other  data  as  an  average  initial  score  in  typewriting. 

Of  the  eighty-three  subjects  who  took  the  tests  thirty-two 
were  eliminated,  leaving  fifty-one  subjects  for  the  major  study. 
The  causes  of  elimination  are  indicated  in  the  following  table : 

Original  size  of  group 83 

Irregular  attendance  20 

Unusually  irregular  performance       3 
Apparent    linearity    of    learning 

curve 5 

Delayed  positive  acceleration ....       2 

Demonstrable  plateau   2 

Total  eliminated 32 

Size  of  group  for  major  study. ..  51 


THE  LEARNING  CURVE  EQUATION  27 

The  twenty  subjects  eliminated  from  the  major  study  on  ac- 
count of  irregular  attendance  are  not  of  interest  in  this  connec- 
tion. Three  subjects  were  eliminated  for  extremely  erratic 
performance  in  the  tests.  It  is  impossible  that  their  real  type- 
writing ability  is  even  approximately  represented  by  their  erratic 
scores.  The  cause  for  their  variability  is  undoubtedly  due  to 
lack  of  consistent  interest  in  their  work  and  in  the  tests.  Most 
of  the  subjects  took  a  competitive  attitude  toward  the  tests,  the 
results  of  which  were  given  them  weekly  by  their  instructor. 

Ten  subjects  had  learning  curves  which  deviated  from  the 
typical  hyperbolic  form  which  we  are  here  considering.  This 
is  a  limitation  of  our  method  which  is  only  applicable  to  the 
hyperbolic  form  of  the  speed-amount  curve.  Of  these  ten  sub- 
jects five  were  eliminated  from  the  major  study  on  account  of 
apparent  linearity  of  the  learning  curves.  No  learning  curve 
can  ever  be  continuously  linear  if  it  is  plotted  in  the  speed* 
amount  form.  If  it  were  linear  the  subject  would  have  no 
physiological  limit  and  he  would  in  time  reach  the  rather  en- 
viable attainment  of  infinite  writing  speed,  which  is  of  course 
absurd.  Another  alternative  with  a  linear  learning  curve  is  that 
it  is  linear  until  it  reaches  the  practice  limit  after  which  it  re- 
mains at  the  limit.  I  can  not  entertain  this  as  a  possibility  for 
it  is  inconceivable  that  an  organic  function  like  learning  pro- 
ceeds according  to  a  linear  relation  until  it  bumps  into  some 
inflexible  practice  limit  at  which  it  stops  and  remains.  The  only 
possible  explanation  of  apparently  linear  learning  curves  that 
I  am  willing  to  entertain  is  that  they  are  in  reality  curved  but 
that  the  degree  of  curvature  is  so  small  that  it  is  concealed  by 
the  variability  of  the  individual  observations.  Such  learning 
curves  are  therefore  indeterminate  unless  they  be  continued  far 
enough  to  make  the  curvature  appear  in  spite  of  the  variations 
of  the  individual  observations.  This  leads  to  the  conclusion 
that  the  accuracy  with  which  the  learning  coefficients  can  be 
determined  is  contingent  on  two  principal  factors.  It  varies 

a)  with  the  degree  of  curvature  of  the  learning  curve,  and 

b)  inversely  as  the  variability  of  the  individual  measurements. 
The  coefficients  of  a  learning  curve  with  minimum  variability 


28  L.  L.  THURSTONE 

may  be  determined  with  a  minimum  amount  of  visible  curvature. 
The  more  variable  the  measurements  the  greater  is  the  degree 
of  curvature  necessary  for  a  fairly  accurate  determination  of 
th  learning  curve  coefficients.  Whether  the  linearity  of  the 
curves  of  these  five  subjects  is  apparent  or  real  can  not  be  settled 
with  the  available  data.  If  the  linearity  is  real  it  constitutes  a 
limitation  in  the  use  of  the  learning  curve  equation. 

Two  subjects  were  eliminated  on  account  of  delayed  positive 
acceleration.  Their  learning  curves  constitute  deviations  from 
the  usual  shape  of  curve  and  can  not  be  handled  by  the  methods 
which  we  are  discussing  here.  It  is  not  certain  that  these  meas- 
ures are  not  simply  cases  of  erratic  performance. 

Two  out  of  the  eighty-three  subjects  showed  clear  evidence 
of  a  plateau.  Whether  this  is  psychologically  significant  or 
simply  due  to  the  fact  that  these  subjects  were  offered  positions 
after  attaining  a  specified  typewriter  proficiency  is  indetermina- 
ble. The  higher  order  learning  curve  which  followed  the  first 
curve  is  not  carried  far  enough  with  either  of  these  two  sub- 
jects to  justify  determining  the  learning  coefficients  for  the  first 
and  second  order  curves. 

We  have  eliminated  twelve  out  of  sixty-three  complete  rec- 
ords. Generalizing  from  this  fact  we  may  conclude  that  the 
speed-amount  form  of  learning  for  typewriting  takes  the  hyper- 
bolic form  in  about  four  cases  out  of  five.  This  justifies  our 
reference  to  it  as  the  typical  but  not  as  the  universal  form  of 
learning  curve. 

2)  THE  COORDINATES  OF  CURVES  FOR  TYPEWRITING 

My  first  intention  was  to  plot  the  learning  curves  with  speed 
as  ordinates  and  time  in  weeks  as  abscissae,  the  speed-time 
form.  Finding  that  the  industry  of  the  subjects  during  the 
practice  hours  varied  immensely  I  decided  that  the  psychological 
analysis  would  be  more  equitable  if  I  measured  practice  in  terms 
of  total  number  of  pages  written  rather  than  in  terms  of  time, 
although  time  is  statistically  more  readily  obtained  than  the 
amount  of  practice.  The  practice  sheets  were  all  turned  in  to 
the  teacher  in  charge  who  tabulated  every  week  the  number  of 


THE  LEARNING  CURVE  EQUATION  29 

pages  written  by  each  subject.  According  to  the  typewriter 
championship  rules,  attainment  should  be  scored  by  deducing 
five  words  from  the  speed  total  for  every  error.  For  the  pur- 
pose of  psychological  study  I  have  separated  errors  from  speed. 
The  learning  curves  are  all  plotted  as  speed  (words  in  four  min- 
utes, disregarding  errors)  against  formal  practice  (pages  writ- 
ten since  entering  the  course).  While  errors  are  entirely  dis- 
regarded in  these  curves  the  subjects  were  of  course  not  informed 
on  this  point.  The  errors  are  studied  separately  by  correlating 
them  with  the  other  learning  characteristics.  In  this  manner 
we  shall  arrive  at  a  statement  of  the  relationship  between  the 
several  learning  characteristics  without  artificially  loading  them 
with  each  other,  as  would  be  the  case  if  we  penalized  the  score 
for  speed  by  the  number  of  errors. 

3)  THE  LEARNING  COEFFICIENTS  FOR  TYPEWRITING 

We  shall  use  the  following  notation  in  studying  typewriter 
learning. 

X  =  Practice,  in  terms  of  the  total  number  of  pages  written 

since  entering  the  course. 
Y  =  Attainment,  in  terms  of  the  number  of  words  written 

in  four  minutes. 
x  =  Number  of  pages  written  when  an  individual  test  is 

taken. 
y  =  Number  of  words  written  in  a  four  minute  test.    It  is 

the  observed  speed  whereas  Y  is  the  speed  indicated  by 

the  learning  curve  equation, 
n  =  Number  of  tests  taken. 
ya  =  Average  speed  in  all  tests  or 
^    Sy 

n 

y20=  Average  speed  after  twenty  weeks  of  practice,  and  sim- 
ilar notation  for  the  average  speed  at  other  stages  of 
learning. 

L  ==  Predicted  practice  limit  in  terms  of  words  written  in 
four  minutes. 

R  =  The  rate  of  learning,  a  constant  which  is  numerically 
large  for  a  low  rate  of  approach  and  numerically  low 
for  a  rapid  rate  of  approach. 

P  =  Equivalent  previous  practice  in  terms  of  practice  units 


30  L.  L.  THURSTONE 

(pages  written).     It  is  the  negative  x-intercept  of  the 
speed-amount  curve. 

d  =  Absolute  deviation  which  expresses  the  deviation  of  any 
single  observation  above  or  below  the  value  indicated 
by  the  learning  curve  at  that  stage  of  learning.  It  is 
positive  when  the  speed  of  any  single  test  is  above  that 
indicated  by  the  learning  curve,  and  negative  when  the 
actual  speed  is  below  the  learning  curve.  It  can  also 
be  represented  by  the  relation 
d  =  y  —  Y 

D  =  Average  deviation  for  all  tests  during  the  year,  or 

2d 

D==  - 
n 

dr  =  Relative  deviation  of  an  individual  test,  determined  by 
the  ratio 


Y 

Dr  =  Average  relative  deviation,  determined  by  the  ratio 


n 

V  =  Coefficient  of  variability,  determined  by  the  ratio 
D 

ya 

e  =  Number  of  errors  made  in  an  individual  test  as  deter- 

mined by  the  Typewriter  Championship  rules. 
E  =  Average  number  of  errors  in  all  tests,  or 


n 
er  =  Relative  inaccuracy  of  an  individual  test,  or 

e 
er  =  — 

y 

Er  =  Average  relative  inaccuracy  for  all  tests,  or 


n 
A  =  Coefficient  of  inaccuracy,  determined  by  the  ratio 

E 
A  =  - 


THE  LEARNING  CURVE  EQUATION  31 

4)  FINDINGS 

a)   Writing  Speed 

Figure  5  indicates  the  relation  between  average  speed  and 
number  of  pages  practiced.  The  law  of  diminishing  returns  is 
shown  by  the  continuity  of  the  points  representing  average  speed 
but  we  can  not  assert  the  universality  of  this  form  because  we 
have  already  eliminated  12  out  of  the  63  available  records.  It 
will  serve  well  as  a  norm  of  average  performance  for  groups 
comparable  with  the  one  here  represented.  It  is  interesting  to 
note  that  the  curve  does  not  pass  through  the  origin.  This  is 
explainable  by  the  fact  that  a  person  who  has  never  written  on 
a  typewriter  can,  nevertheless,  even  on  the  first  trial,  make  some 
finite  score.  I  could  not  readily  obtain  a  test  from  the  fifty-one 
subjects  prior  to  formal  instruction  on  the  typewriter.  In  order 
to  ascertain  what  a  truly  initial  score  is,  I  asked  ten  of  my  stu- 
dents who  had  never  touched  a  typewriter  to  take  a  four  minute 
test.  The  mean  as  well  as  the  average  of  these  ten  subjects  was 
27  words  in  four  minutes.  This  is  the  point  to  which  the  compo- 
site learning  curve  in  figure  5  projects. 

The  solid  line  of  figure  5  represents  the  hyperbolic  curve  form. 
It  is  a  good  fit  on  the  data  which  are  represented  by  the  small 
circles.  The  equation  of  the  composite  curve  is 

2i6.(X+i9.) 
Y  = 40 

X+I48. 

in  which  Y  is  the  average  score  for  the  51  subjects  and  X  is  the 
number  of  pages  of  practice.  The  predicted  limit  of  practice 
for  speed  is  216.  words  in  four  minutes,  which  agrees  well  with 
average  typewriter  speed.  The  average  of  the  limits  of  practice 
as  determined  from  the  individual  curves  is  214.  The  equivalent 
previous  practice  for  the  composite  curve  is  19.  This  indicates 
that  the  general  experience  which  these  subjects  brought  to  their 
first  practice  on  the  typewriter  was  equivalent  to  nineteen  pages 
of  formal  practice.  The  rate  of  learning,  R,  for  the  composite 
curve  is  129,  a  constant  which  varies  inversely  with  the  relative 
rapidity  with  which  the  limit  of  practice  is  approached.  The 
average  of  the  rates  of  approach  as  determined  from  the  indi- 
vidual curves  is  137.  The  composite  curve,  figure  13,  is  plotted 


32  L.  L.  THURSTONE 

against  time  in  weeks  instead  of  against  amount  of  practice.  It 
has  an  entirely  different  appearance.  It  shows  the  initial  posi- 
tive acceleration  previously  discussed. 

We  shall  now  turn  to  the  individual  records  and  ascertain  by 
correlation  methods  the  interrelations  of  the  learning  character- 
istics for  typewriting. 

The  correlation  between  the  predicted  limit  of  practice  and 
the  average  speed  of  writing  for  all  tests  during  the  year  is 
0.68.  This  correlation  could  not  even  theoretically  be  close  to 
unity  for  while  the  fast  writers  tend  to  approach  the  higher 
practice  limits  there  is  considerable  variation  in  the  rate  at  which 
the  limit  is  approached.  It  should  be  noted  that  the  predicted 
limit  was  not  attained  by  these  subjects.  The  limit  is  predicted 
on  the  basis  of  the  curve  shape.  If  the  predicted  limit  used  here 
agrees  at  all  closely  with  the  ultimate  writing  speed  of  these  sub- 
jects, the  correlation  of  0.68  between  the  practice  limit  and  the 
average  writing  speed  during  eight  months'  instruction  indicates 
that  the  latter  measure  is  not  a  very  reliable  criterion  of  ultimate 
proficiency.  I  think  that  it  constitutes  another  piece  of  evidence 
against  hasty  and  self-confident  predictions  based  on  so  called 
vocational  mental  tests. 

The  correlation  between  the  predicted  limit  of  practice,  L, 
and  the  rate  of  learning,  R,  is  0.75.  This  indicates  that  those 
who  approach  a  high  limit  of  practice  in  speed  generally  ap- 
proach their  limit  at  a  lower  rate  than  those  who  are  approach- 
ing a  low  speed  as  their  limit.  The  regression  is  linear  and  hence 
the  converse  is  true,  namely  that  those  who  approach  a  low  limit 
of  practice  generally  approach  their  limit  at  a  relatively  higher 
rate  than  those  who  have  a  high  limit  of  practice. 

The  correlation  between  predicted  limit  of  writing  speed  and 
average  number  of  errors  in  unit  time  is  o.io.  Hence  we  con- 
clude there  is  no  discernible  relation  between  the  normal  writing 
speed  and  the  absolute  number  of  errors  made  per  unit  time. 
But  there  is  a  noticeable  relation  between  the  coefficient  of  in- 
accuracy, A,  and  the  practice  limit,  L.  The  coefficient  of  corre- 
lation is  only  — 0.16  but  the  regression  is  non-linear.  Those 
who  approach  a  low  practice  limit  for  writing  speed  tend  to  be 
inaccurate,  but  those  who  approach  a  high  practice  limit  are 


THE  LEARNING  CURVE  EQUATION  33 

either  accurate  or  inaccurate.  On  the  other  hand  those  who  are 
unusually  accurate  tend  to  be  fast  writers  whereas  those  who 
are  inaccurate  are  either  fast  or  slow  writers.  The  number  of 
subjects  is  not  large  enough  to  warrant  the  calculation  of  the 
eta  coefficient. 

The  correlation  between  the  predicted  writing  speed  and  the 
speed  after  eight  weeks  of  practice  is  0.27.  Making  use  of  the 
actual  data  instead  of  predicted  performance,  we  find  that  the 
correlation  between  the  speed  after  eight  weeks  practice  and 
that  after  twenty  weeks  practice  is  0.74.  The  correlation  be- 
tween predicted  limit  and  the  average  writing  speed  for  all  tests 
is  0.68. 

In  this  connection  I  wish  to  suggest  what  will  perhaps  be  a 
more  reliable  technique  in  psychological  prognosis.  If  a  high 
degree  of  relationship  can  be  established  between  the  learning 
curve  constants  for  a  complex  function,  performance  in  which 
is  to  be  predicted,  and  the  corresponding  constants  for  a  simple 
learning  function  which  can  be  completed  during  a  single  sitting, 
then  the  constants  for  the  simple  learning  test  would  have  diag- 
nostic value  in  predicting  performance  in  the  complex  function. 
Such  coefficients  would  not  be  subject  to  the  accidents  of  a  first 
performance  but  would  represent  the  organically  more  significant 
learning  function  as  such.  The  diagnostic  value  of  such  a  tech- 
nique is  largely  dependent  on  the  degree  of  difficulty  of  the 
material  to  be  learned. 
b)  The  errors 

The  relation  between  the  average  absolute  number  of  errors 
made  in  each  test  and  the  number  of  weeks  of  practice  is  indi- 
cated in  figure  16.  This  shows  that  the  number  of  errors  in  unit 
time  increases  with  practice.  The  relation  between  these  two 
attributes  may  be  expressed  by  the  empirical  equation 

e  =  o.i2T  +  2.i  66 

in  which  e  is  the  average  number  of  errors  in  a  four  minute 
test,  and  T  is  the  number  of  weeks  of  instruction.  The  equation 
expresses  a  norm  of  average  performance. 

The  analogous  relation  between  average  absolute  number  of 
errors  and  writing  speed  during  the  year  is  shown  in  figure  17. 


34  L,  L.  THURSTONE 

This  also  indicates  that  the  number  of  errors  made  in  unit  time 
increases  with  the  attainment  of  writing  speed.  The  relation 
may  be  expressed  by  the  empirical  equation 

e  =  o.023y  +1.5  67 

which  is  fairly  representative  within  the  limits  of  observation. 
However,  the  relative  inaccuracy  decreases  with  practice  as  in- 
dicated in  figure  18,  i.e.,  the  number  of  errors  per  page  decreases 
with  practice  but  the  number  of  errors  per  unit  time  increases 
with  practice.  There  is  no  discernible  relation  between  the  rela- 
tive inaccuracy  (errors  per  unit  time)  and  the  rate  of  learning. 
c)  The  variability 

Those  who  have  a  high  practice  limit  for  writing  speed  usually 
have  a  larger  average  deviation  from  their  learning  curves  than 
those  who  approach  a  low  practice  limit.  This  statement  must 
be  considered  in  connection  with  relation  between  predicted  limit 
of  writing  speed  and  the  average  relative  deviation.  The  corre- 
lation between  predicted  writing  speed  and  average  relative. de- 
viation for  all  tests  is  0.27,  indicating  that  the  fast  writers  have 
a  slight  tendency  to  be  more  erratic  in  speed  than  the  slow  writers, 
even  though  the  measure  of  variability  is  taken  as  the  ratio  of 
deviation  to  writing  speed.  According  to  this  measure  the  writer 
of  60  words  per  minute  is  allowed  a  deviation  from  his  repre- 
sentative learning  curve  twice  that  of  a  writer  of  30  words  per 
minute.  But  even  according  to  this  relative  standard  of  varia- 
bility the  fast  writers  tenc}  to  be  slightly  more  erratic  in  speed. 

Figure  14  indicates  that  the  deviations  from  the  learning  curve 
increase  with  practice  but  figure  15  shows  that  the  ratio  of  devia- 
tion to  theoretical  \vriting  speed,  as  indicated  by  the  curve,  de- 
creases with  practice.  It  is  apparent  that,  just  as  one  would 
expect,  the  absolute  deviations  increase  with  practice  but  the 
relative  deviations  decrease  with  practice.  The  decrease  in  the 
relative  deviation  with  practice  does  not  become  noticeable  until 
after  about  three  months  but  after  that  the  relation  is  approxi- 
mately linear  with  practice  time.  In  other  words,  the  variability 
of  the  writing  speed  for  any  individual  subject  tends  to  decrease 
with  practice,  but  if  he  is  a  fast  writer  he  tends  to  be  more 
Nvariable  in  his  writing  speed  than  if  he  is  a  slow  writer,  even 
wiaen  the  variability  is  measured  in  relative  terms. 


SUMMARY 
i )  FORMS  OF  THE  LEARNING  CURVE 

We  have  discussed  four  different  forms  in  whidi  most  learn- 
ing data  can  be  graphed.  We  have  called  these  forms  i)  the 
speed-amount  curve,  2)  the  speed-time  curve,  3)  the  time-time 
curve,  and  4)  the  time-amount  curve. 

The  speed-amount  curve  is  plotted  as  speed,  number  of  suc- 
cessful acts  per  unit  time,  or  a  multiple  thereof,  against  the  total 
number  of  formal  practice  acts,  or  some  multiple  of  it.  In  plot- 
ting typewriter  learning  we  have  used  words  in  four  minutes, 
and  the  total  number  of  pages  written  as  the  coordinates  of  the 
speed-amount  curve.  This  form  of  curve  is  illustrated  in  figure 
5  which  gives  average  writing  speed  for  fifty-one  subjects  against 
total  number  of  pages  written.  The  small  circles  indicate  the 
observations  and  the  solid  line  indicates  the  general  trend  of  the 
learning.  The  solid  line  is  represented  by  the  general  equation 

L(X+P) 

Y  = 20 

(X+P)+R 
in  which 

L  =  Predicted  practice  limit  in  terms  of  speed  units. 
X  =  Pages  written. 

Y  =  Writing  speed  in  terms  of  words  in  four  minutes. 
P  =  Equivalent  previous  practice  in  terms  of  pages. 
R  =  Rate  of  learning,  a  constant  which  varies  inversely  as 
the  relative  rapidity  with  which  the  practice  limit  is  being 
approached. 

K  =  P  +  K,  a  constant  used  for  convenience. 
The  particular  line  of  figure  5  is  represented  by  the  equation 
Y  =  2i6.(X+i9.) 

X+i48. 

which  we  may  interpret  as  follows.  The  predicted  average  prac- 
tice limit,  L,  for  the  group  of  fifty-one  subjects  is  216  words 
in  four  minutes  or  about  54  words  a  minute.  The  rate  of  learn- 
ing, R,  is  a  constant  which  in  this  curve  has  the  value  of  129. 


36  L.  L.  THURSTONE 

Its  only  usefulness  is  in  comparing  the  rates  of  several  learning 
curves  with  each  other.  By  itself,  and  for  a  single  curve,  it  has 
no  significance.  When  used  to  compare  several  learning  curves 
the  precaution  must  be  observed  that  all  curves  so  compared  be 
plotted  by  the  same  units  for  the  coordinates.  The  equivalent 
previous  practice  is  nineteen  pages.  This  is  interpreted  to  mean 
that  the  general  experience  which  these  students  brought  to  their 
first  instruction  on  the  typewriter  was  equivalent  to  writing  nine- 
teen pages  on  the  machine.  This  coefficient  as  well  as  the  pre- 
dicted limit  is  based  on  the  assumption  that  the  unmeasured  part 
of  the  learning  before  and  after  the  observations  followed  the 
hyperbolic  law.  This  assumption  seems  to  be  fairly  safe  since 
other  learning  curves,  the  actual  observations  for  which  start 
with  practically  zero  attainment  and  continue  almost  to  the  prac- 
tice limit,  usually  follow  the  hyperbolic  form.  See  figure  n 
which  represents  a  substitution  test  learning  curve  carried  almost 
to  the  practice  limit,  and  the  curve  for  subject  No.  23  in  figure  3 
for  typewriter  learning  which  projects  to  the  origin. 

The  time-amount  curve  is  plotted  as  time,  t,  per  unit  amount 
of  work  against  number  of  formal  practice  acts,  X.  Its  equa- 
tion is 

X+K 

t  = 49 

L(X+P) 

with  notation  similar  to  that  of  equation  20.  Learning  data  can 
be  changed  from  the  time-amount  form  into  the  speed-amount 
form  and  vice  versa  by  noting  the  fact  that  speed,  Y,  is  propor- 
tional to  the  reciprocal  of  the  time,  t,  per  unit  amount  of  work. 

The  time-time  curve  is  plotted  as  time,  t,  per  unit  amount  of 
work  against  total  practice  time,  T.  Its  equation,  54,  is  derived 
from  equation  20.  This  equation  is  too  cumbersome  for  prac- 
tical work  and  the  speed-amount  or  time-amount  curves  should 
therefore  be  used  unless  one  adopts  a  simple  empirical  equation 
for  the  time-time  form. 

The  speed-time  curve  is  plotted  as  speed,  Y,  against  total  prac- 
tice time,  T.  Its  equation,  55,  is  too  unwieldy  for  practical  work 
but  it  serves  to  demonstrate  the  following  proposition  regarding 


THE  LEARNING  CURVE  EQUATION  37 

positive  acceleration.  If  we  assume  that  the  typical  form  of 
speed-amount  curve  is  hyperbolic,  then  the  learning  curve  in 
the  speed-time  form  must  necessarily  have  an  initial  positive 
acceleration  which  changes  to  a  negative  acceleration  when  the 
attainment  has  reached  one-third  of  the  limit  of  practice.  The 
positive  acceleration  can  not,  of  course,  be  obtained  when  the 
initial  score  is  greater  than  one-third  of  the  practice  limit.  It 
can  only  be  observed  when  the  initial  score  is  less  than  one-third 
of  the  practice  limit. 

The  influence  of  different  values  for  the  learning  coefficients 
on  the  shape  of  the  learning  curve  may  be  summarized  in  the 
following  comparisons.  Figure  7  shows  two  learning  curves 
approaching  two  different  limits  at  the  same  rate.  Figure  8 
shows  two  curves  approaching  the  same  limit  at  different  rates. 
Figure  9  shows  two  curves  approaching  the  same  limit  at  the 
same  rate  but  differing  in  the  amount  of  previous  practice. 
Curve  A  has  a  start  of  forty  practice  units  over  curve  B.  Fig- 
ure 10  shows  two  curves,  one  approaching  a  high  limit  at  a  low 
rate,  the  other  approaching  a  lower  limit  at  a  high  rate.  The 
important  feature  of  this  comparison  is  that  one  who  learns 
rapidly  but  with  a  low  limit  will  do  better  in  the  first  stages  of 
the  learning  than  one  who  learns  slowly  with  a  high  limit.  The 
comparison  shifts  later  in  favor  of  the  learner  with  the  high 
limit.  This  is  a  condition  which  experimenters  on  learning 
should  be  on  the  look-out  for  in  order  to  guard  against  the  er- 
roneous comparison  of  two  subjects  from  insufficient  practice 
data. 

2)  OUTLINE  FOR  CALCULATING  THE  LEARNING  COEFFICIENTS 
We  shall  describe  two  methods  of  calculating  the  coefficients. 
These  are  i)  the  method  of  all  points,  and  2)  the  method  of 
three  points. 

i )  Method  of  all  points: 

i )  Arrange  the  data  in  two  columns  as  follows :  X,  the  total 
number  of  formal  practice  acts  since  the  beginning  of  practice, 
or  a  multiple  of  this  number,  and  Y,  the  number  of  successful 
acts  in  unit  time,  or  a  multiple  of  this  number.  The  multiple 


38  L.  L.  THURSTONE 

used  for  the  Y-column  need  not  of  course  be  the  same  as  that 
for  the  X-column. 

2)  Draw  a  chart  analogous  to  figure  5.     Leave  room  for  a 
negative    x-intercept.      Always  include  the  zero  point  of  the 
y-scale  on  the  chart. 

3 )  Select  a  representative  initial  score.  In  figure  5  the  actually 
observed  initial  score,  27,  was  used.     Denote  this  by  the  sym- 
bol Yi. 

4)  Compute  the  values  of  X/(Y— Yx)   and  XY/(Y— Y±) 
for  each  observation.    Arrange  these  in  two  columns. 

5)  Draw  a  chart  analogous  to  figure  6  with  the  coordinates 
determined  in  step  4.     If  the  data  so  plotted  fall  nearly  on  a 
straight  line  the  speed-amount  curve  is  hyperbolic.     If  it  does 
not,  the  use  of  the  learning  curve  equation  is  not  justified  and 
other  methods  must  be  resorted  to. 

6)  Fit  a  straight  line  through  these  points  in  figure  6  in  such 
a  manner  that  there  are  about  as  many  points  on  one  side  of 
the  line  as  there  are  points  on  the  other  side.    This  procedure 
is  called  "rectifying  the  equation."    The  line  can  be  fitted  more 
accurately  by  the  method  of  least  squares  but  since  that  is  a 
rather  laborious  procedure  it  should  be  avoided  unless  the  points 
are  so  badly  scattered  that  they  can  not  readily  be  fitted  by  in- 
spection.    Even  then  it  is  doubtful  whether  one  is  justified  in 
applying  the  equation  to  learning  data  so  erratic  that  a  straight 
line  can  not  be  fitted  by  inspection. 

7)  Continue  this  line  until  it  intersects  the  x-axis.     The  x- 
intercept  gives  the  value  of  K/Y±,  and  since  the  value  of  Yx  is 
already  known  the  value  of  K  can  be  readily  determined. 

8)  The  slope  of  the  line  is  numerically  equal  to  the  predicted 
limit,  L. 

9)  The  constant  P  may  then  be  determined  from  the  equation 

p  =  K-Y* 
L 

10)  The  constant  R  is  then  determined  by  the  equation 

R  =  K  — P 
since  K  and  P  are  known. 


THE  LEARNING  CURVE  EQUATION  39 

2)  Method  of  three  equidistant  points. 

The  first  two  steps  of  this  method  are  identical  with  the  first 
two  steps  in  the  method  of  all  points. 

3)  Draw  a  smooth  curve  through  the  observations.     If  the 
data  show  irregularities  in  the  rate  of  learning  draw  the  smooth 
curve  so  that  it  has  approximately  as  many  observations  above 
the  line  as  there  are  observations  below  the  line.    A  ragged  line 
through  all  the  more  or  less  erratic  observations  will  not  serve 
the  purpose.    If  the  smooth  curve  representative  of  the  data  is 
not  of  the  hyperbolic  form  the  method  of  this  learning  curve 
equation  is  not  applicable. 

4)  Select  the  three  following  points: 

X,;*, 

in  which  Xt  is  zero,  and  Y±  is  the  ordinate  to  the  smooth  curve 
at  this  value  of  X.  If  the  learning  data  have  no  irregularities 
the  value  of  Yx  will  be  identical  with  the  initial  score. 

X2;  Y2 

in  which  X2  is  one  half  of  the  total  amount  of  formal  practice 
and  Y2  is  the  representative  ordinate  to  the  curve  for  this  value 
of  X. 

X3;Y3 

in  which  X3  is  the  total  amount  of  formal  practice  and  Y3  is  the 
ordinate  to  the  smooth  curve  at  this  value  of  X.  If  the  learning 
data  show  no  irregularities,  this  will  be  identical  with  the  final 
score. 

5)  Determine  the  numerical  value  of  the  constant  K  from 
the  equation: 

X3(Y2-Y3) 

1  Y8+Y1-2Y2 

6)  Determine  the  numerical  value  of  constant  L  by  means 
of  the  following  equation: 

Y8(X3+K)-Y1.K 

J-/   ~— ~ ~~ ~~~~~~ — — — ^-^— — — — — 

X3 

7)  Determine  the  numerical  value  of  the  constant  P  by  the 
following  equation: 


40  L.  L.  THURSTONE 

8)  Determine  the  numerical  value  of  the  constant  R  by  the 
following  equation : 

R  =  K  —  P 

After  the  constants  of  the  learning  curve  equation  have  been 
numerically  evaluated  it  is  best  to  check  the  arithmetical  work 
by  computing  the  theoretical  value  of  the  attainment  for  one 
or  two  points  according  to  the  following  formula.  These  the- 
oretical values  of  attainment  should  not  differ  much  from  the 
actually  observed  values  unless  the  learning  has  been  very  erratic. 

y  =  L(X+P) 

(X+P)+R 

A  gross  measure  of  the  variability  of  the  learning  may  be 
determined  from  the  equation 

D 
y  — 

7a 

in  which  D  is  the  average  deviation  from  the  theoretical  curve 
for  all  the  observations,  and  ya  is  the  average  attainment  as 
determined  from  all  observations. 
3)  TYPEWRITER  LEARNING 

The  following  relations  were  found  to  be  significant  with  re- 
gard to  typewriter  learning. 

The  correlation  between  the  predicted  practice  limit  and  the 
average  writing  speed  for  all  tests  which  covered  about  seven 
months  is  -j-o.68.  The  correlation  between  practice  limit  and 
rate  of  learning  is  +0.75.  This  indicates  that  those  who  ap- 
proach a  high  practice  limit  usually  do  so  at  a  lower  rate  than 
those  who  approach  a  low  limit  since  the  coefficient,  R,  for  the 
rate  of  learning  varies  inversely  with  the  rate  of  learning.  There 
is  a  noticeable  relation  between  accuracy  and  the  predicted  prac- 
tice limit.  Those  who  approach  a  low  practice  limit  for  writing 
speed  are  usually  inaccurate,  but  those  who  approach  a  high 
practice  limit  are  either  accurate  or  inaccurate.  On  the  other 
hand  those  who  are  inaccurate  are  either  fast  or  slow  writers. 
The  number  of  subjects,  51,  is  not  large  enough  to  warrant  the 
calculation  of  the  eta-coefficient.  The  correlation  between  the 
predicted  practice  limit  and  speed  as  determined  in  the  test  at 


THE  LEARNING  CURVE  EQUATION  41 

the  8th  week  is  +°-27-  There  seems  to  be  no  relation  between 
the  predicted  writing  speed  at  the  limit  of  practice  and  the  num- 
ber of  errors  made  in  unit  time. 

The  number  of  errors  made  in  unit  writing  time  increases 
with  practice.  Similarly  there  is  a  positive  relation  between  the 
number  of  errors  in  unit  time  and  writing  speed  during  practice. 
However,  the  number  of  errors  per  unit  amount  of  work  de- 
creases with  practice.  See  figures  16,  17,  and  18.  There  is  no 
discernible  relation  between  the  relative  accuracy  and  the  rate 
of  learning. 

Those  who  have  a  high  practice  limit  for  writing  speed  usually 
have  larger  relative  deviations  from  their  theoretical  learning 
curves.  The  variability  of  the  writing  speed  for  any  individual 
subject  tends  to  decrease  with  practice,  but  if  the  student  is  a 
fast  writer  he  tends  to  be  more  variable  in  writing  speed,  than 
if  he  is  a  slow  writer,  even  when  the  variability  is  measured  in 
relative  terms.  According  to  this  standard  of  variability  the 
writer  of  60  words  per  minute  is  allowed  a  deviation  from  his 
representative  learning  curve  twice  that  of  a  writer  of  30  words 
per  minute.  But  even  according  to  this  relative  standard  of 
variability  the  fast  writers  tend  to  be  slightly  more  erratic  in 
speed. 

Considerable  ambiguity  in  discussions  about  learning  curves 
has  been  caused  by  the  comparison  of  learning  curves  with  dif- 
ferent units  for  the  coordinates.  Thus  we  are  entirely  safe  in 
saying  that  the  speed-amount  curve  is  never  continuously  linear. 
It  would  lead  to  infinite  speed  of  performance  which  is  of  course 
absurd.  But  while  that  statement  is  obviously  true  it  does  not 
entitle  us  to  jump  to  the  denial  of  say  linear  error-time  curves. 
It  is  quite  possible  for  errors  plotted  against  time  to  be  linear. 
Therefore  we  should  always  specify  the  coordinates  for  the 
curves  we  are  discussing. 

While  I  have  confined  myself  throughout  to  what  I  have  called 
the  typical  hyperbolic  form  of  the  speed-amount  curve  it  is  quite 
essential  to  keep  in  mind  that  this  form  of  curve  is  not  universal 
and  that  consequently  it  is  impossible  to  make  sweeping  gen- 
eralizations except  in  so  far  as  we  explicitly  limit  ourselves  to 


42  L.  L.  THURSTONE 

the  relations  which  follow  from  the  assumed  hyperbolic  form 
with  which  we  started. 

The  preceding  pages  have  been  filled  with  so  much  algebraic 
manipulation  that  the  reader  who  has  long  since  dropped  the 
algebraic  thinking  of  his  school  days  may  find  their  very  appear- 
ance formidable  and  distasteful.  For  the  benefit  of  those  who 
have  acquired  an  aversion  against  symbolic  notation  I  wish  to 
call  attention  to  the  outline  for  calculating  the  coefficients  and 
the  section  on  learning  curve  forms  in  the  summary.  In  those 
sections  will  be  found  all  that  is  really  essential  in  applying  the 
method.  If  the  use  of  empirical  equations  in  the  quantitative 
study  of  the  multifarious  aspects  of  memory  is  at  all  furthered 
by  the  present  study  I  shall  be  content  though  the  particular 
forms  used  here  are  superseded  by  others. 


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APR    9  19K 


^sr 


i -Eft  3.3  y; 


LD 


1961 


JAN  14  m 


F?  b  b 


HAY  1  6  1961 


REC'D 


LD  21-100m-7,'33 


YC  63549 


405950 


*  /; 
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